L(s) = 1 | + (2.28 + 0.159i)2-s + (−0.0468 − 1.34i)3-s + (3.19 + 0.449i)4-s + (1.25 + 1.85i)5-s + (0.107 − 3.07i)6-s + (−0.711 − 1.23i)7-s + (2.75 + 0.584i)8-s + (1.19 − 0.0833i)9-s + (2.56 + 4.42i)10-s + (0.354 + 3.37i)11-s + (0.453 − 4.31i)12-s + (−1.02 − 1.51i)13-s + (−1.42 − 2.92i)14-s + (2.42 − 1.76i)15-s + (−0.0251 − 0.00722i)16-s + (1.08 − 2.68i)17-s + ⋯ |
L(s) = 1 | + (1.61 + 0.112i)2-s + (−0.0270 − 0.775i)3-s + (1.59 + 0.224i)4-s + (0.559 + 0.828i)5-s + (0.0437 − 1.25i)6-s + (−0.268 − 0.465i)7-s + (0.972 + 0.206i)8-s + (0.397 − 0.0277i)9-s + (0.809 + 1.39i)10-s + (0.106 + 1.01i)11-s + (0.130 − 1.24i)12-s + (−0.284 − 0.421i)13-s + (−0.381 − 0.781i)14-s + (0.627 − 0.456i)15-s + (−0.00629 − 0.00180i)16-s + (0.263 − 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.47482 - 0.371451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.47482 - 0.371451i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.25 - 1.85i)T \) |
| 19 | \( 1 + (3.97 + 1.79i)T \) |
good | 2 | \( 1 + (-2.28 - 0.159i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (0.0468 + 1.34i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (0.711 + 1.23i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.354 - 3.37i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.02 + 1.51i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.08 + 2.68i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-4.05 - 3.91i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (3.39 + 8.41i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (5.94 - 6.59i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-2.89 - 2.10i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.13 + 1.75i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (1.51 - 8.59i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 3.95i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-1.34 - 0.189i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (-0.247 - 0.991i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-6.69 - 6.46i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-0.999 - 0.624i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (13.4 + 7.13i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (-4.75 + 7.05i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.502 - 14.3i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (-9.16 + 10.1i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (4.73 - 1.35i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (2.06 - 1.29i)T + (42.5 - 87.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.31821602367511322579929824995, −10.27875394044615557282350084988, −9.434053396177518766705075713873, −7.51370022708161467538502323245, −7.04162804927541937203789923737, −6.36235214549135286308727337452, −5.29154899878294317520645575341, −4.22108945746239552015614273241, −3.03422243177028876144601881114, −1.95376689176368109709995436318,
2.01417744868746875625158696204, 3.48852086056138177624385272176, 4.28812135632443673789511194551, 5.25895084292903962630833490214, 5.85331809350486419010502782659, 6.88541519890301677779056384566, 8.604741671191958340236208417062, 9.247915696602287729029226554015, 10.41206643953036724717172367839, 11.14771995184991789874739303312